Quantum noise process analysis method and apparatus, device, and storage medium

ABSTRACT

This disclosure describes a quantum noise process analysis method, device, and storage medium, in the field of quantum processing technologies. The method may include performing quantum process tomography (QPT) on a quantum noise process of a target quantum system, to obtain dynamical maps of the quantum noise process, wherein the QPT involves at least one measurement of the target quantum. The method further includes extracting transfer tensor maps (TTMs) of the quantum noise process from the dynamical maps; and analyzing the quantum noise process according to the TTMs. The TTM is used for representing a dynamical evolution of the quantum noise process to reflect the law of evolution of the dynamical maps of the quantum noise process over time. As a result, richer and more comprehensive information about the quantum noise process can be obtained by analyzing the quantum noise process based on the TTM of the quantum noise process than by pure QPT, thereby achieving a more accurate and more comprehensive analysis of the quantum noise process.

RELATED APPLICATION

This application is a continuation of and claims priority to the PCTInternational Application No. PCT/CN2020/084897, filed with the NationalIntellectual Property Administration, PRC on Apr. 15, 2020, which claimspriority to Chinese Patent Application No. 201910390722.5, filed withthe National Intellectual Property Administration, PRC on May 10, 2019,both entitled “QUANTUM NOISE PROCESS ANALYSIS METHOD AND APPARATUS,DEVICE, AND STORAGE MEDIUM”, which are incorporated herein by referencein their entireties.

FIELD OF THE TECHNOLOGY

Embodiments of this application relate to the field of quantumtechnologies, and in particular, to a quantum noise process analysistechnology.

BACKGROUND OF THE DISCLOSURE

A quantum noise process is a quantum information pollution processcaused by the interaction between a quantum system or quantum devicewith a bath or by the imperfection in quantum control.

In the related art, information about a dynamical map of a quantum noiseprocess is extracted through quantum process tomography (QPT). The QPTis a mathematical description of inputting a group of standard quantumstates to a noise channel and reconstructing a quantum noise processthrough a series of measurement processes.

The limited information about the quantum noise process obtained throughpure QPT is insufficient to accurately and comprehensively analyze thequantum noise process.

SUMMARY

Embodiments of this application provide a quantum noise process analysismethod and apparatus, a device, and a storage medium, to resolve theforegoing technical problem in the related art. The technical solutionsare as follows:

According to an aspect, an embodiment of this application provides aquantum noise process analysis method, including:

performing quantum process tomography (QPT) on a quantum noise processof a target quantum system, to obtain dynamical maps of the quantumnoise process;

extracting transfer tensor maps (TTMs) of the quantum noise process fromthe dynamical maps, the TTMs being used for representing a dynamicalevolution of the quantum noise process; and

analyzing the quantum noise process according to the TTMs.

According to another aspect, an embodiment of this application providesa quantum noise process analysis apparatus, including:

an obtaining module, configured to perform quantum process tomography(QPT) on a quantum noise process of a target quantum system, to obtaindynamical maps of the quantum noise process;

an extraction module, configured to extract TTMs of the quantum noiseprocess from the dynamical maps, the TTMs being used for representing adynamical evolution of the quantum noise process; and

an analysis module, configured to analyze the quantum noise processaccording to the TTMs.

According to still another aspect, an embodiment of this applicationprovides a computer device, including a processor and a memory, thememory storing at least one instruction, at least one program, a codeset or an instruction set, the at least one instruction, the at leastone program, the code set or the instruction set being loaded andexecuted by the processor to implement the foregoing quantum noiseprocess analysis method.

According to yet another aspect, an embodiment of this applicationprovides a computer-readable storage medium, storing at least oneinstruction, at least one program, a code set or an instruction set, theat least one instruction, the at least one program, the code set or theinstruction set being loaded and executed by a processor to implementthe foregoing quantum noise process analysis method.

According to still yet another aspect, an embodiment of this applicationprovides a computer program product, when executed, the computer programproduct being configured to perform the foregoing quantum noise processanalysis method.

The technical solutions provided in the embodiments of this applicationmay include at least the following beneficial effects:

In the technical solutions provided in this application, QPT isperformed on a quantum noise process, to obtain dynamical maps of thequantum noise process, and a TTM of the quantum noise process is furtherextracted from the dynamical maps of the quantum noise process. The TTMis used for representing a dynamical evolution of the quantum noiseprocess, that is, reflecting the law of evolution of the dynamical mapsof the quantum noise process over time. Compared with pure QPT, thisapplication can obtain richer and more comprehensive information aboutthe quantum noise process. Therefore, when the quantum noise process isanalyzed based on the TTM of the quantum noise process, a more accurateand comprehensive analysis of the quantum noise process can be achievedbased on the richer and more comprehensive information.

BRIEF DESCRIPTION OF THE DRAWINGS

To describe the technical solutions of the embodiments of thisapplication more clearly, the following briefly describes theaccompanying drawings for illustrating the embodiments. The accompanyingdrawings in the following description are merely examples, and a personof ordinary skill in the art may still derive other accompanyingdrawings according to these accompanying drawings without creativeefforts.

FIG. 1 is an overall flowchart of a technical solution of thisdisclosure.

FIG. 2 is a flowchart of a quantum noise process analysis methodaccording to an embodiment of this disclosure.

FIG. 3 to FIG. 8 exemplarily show schematic diagrams of several groupsof experimental results in a simulated bath.

FIG. 9 to FIG. 14 exemplarily show schematic diagrams of several groupsof experimental results in a real bath.

FIG. 15 is a block diagram of a quantum noise process analysis apparatusaccording to an embodiment of this disclosure.

FIG. 16 is a block diagram of a quantum noise process analysis apparatusaccording to another embodiment of this disclosure.

FIG. 17 is a schematic structural diagram of a computer device accordingto an embodiment of this disclosure.

DESCRIPTION OF EMBODIMENTS

To make the objectives, technical solutions, and advantages of thisdisclosure clearer, the following further describes implementations ofthis disclosure in detail with reference to the accompanying drawings.

Before the embodiments of this disclosure are described, some termsinvolved in this disclosure are explained first.

1. Quantum system: It is part of the entire universe, and its motion lawfollows the quantum mechanics.

2. Quantum state: All information of the quantum system is representedby a quantum state ρ. ρ is a d×d complex matrix, where d is the quantityof dimensions of the quantum system.

3. Quantum noise process: It is a quantum information pollution processcaused by interaction between the quantum system or quantum device withthe bath or by imperfection in quantum control. Mathematically, thisprocess is a channel represented by using a super-operator, and if theprocess is expanded to a higher dimensionality, the process may berepresented by using a matrix.

4. Memory kernel: It is an operator acting on the quantum state, andincludes all information about system decoherence triggered by the bath.

5. Second-order memory kernel: It is a second-order series expansion ofthe memory kernel in the coupling strength between the quantum systemand the bath.

6. Second-order correlation function of noise: It is a correlationfunction of a system noise between two different time points, and isused for calculating a frequency spectrum of the noise.

7. Transfer tensor map (TTM): It is a map recursively extracted fromdynamical maps of the quantum noise process, and this map encodes thememory kernel of the quantum system, and can be used for predicting adynamical evolution of the quantum system and determining properties ofthe noise.

8. Quantum process tomography (QPT): It is a mathematical description ofinputting a group of standard quantum states to a noise channel andreconstructing a quantum noise process through a series of measurementprocesses.

In quantum information processing, all information of the quantum systemis represented by an evolution ρ(t) of a quantum state over time t. ρ(t)is a d×d complex matrix. Any quantum process, either a quantuminformation processing process or a quantum noise process, may berepresented by using a dynamical map when the system and the bath are ina separable state initially:

ε(ρ)=Σ_(k) A _(k) ρA _(k) ^(†),

where A_(k) is also a d×d matrix and meets Σ_(i)A_(i) ^(†)A≤I, andrepresents a k^(th) component of the influence of the bath on thequantum system, and I is an identity matrix. A_(k) ^(†) represents aHermitian conjugate, that is, a complex conjugate transpose, of A_(k) .Because of completeness of a finite-dimensional complex matrix space, anorthogonal basis matrix set {E_(i)} in a d×d matrix space is defined,and then the following may be obtained:

A _(i)=Σ_(m) a _(im) E _(m),

where a_(im)∈

,

represents a complex set, E_(m) is an element in {E_(i)}, and both i andm are positive integers.

In this way, the following may be obtained:

${{ɛ(\rho)} = {\sum\limits_{m,n}{\chi_{m,n}E_{m}\rho E_{n}^{\dagger}}}},{{{where}\mspace{14mu} \chi_{m,n}} = {\sum\limits_{i}{a_{mi}a_{ni}^{*}}}},$

and is an element of a complex transformation matrix χ whose index is m,n, E_(n) ^(†) represents a Hermitian conjugate of E_(n), E_(n) is anelement in {E_(i)}, and ρ represents an input state.

In the related art, a quantum noise process analysis method based on QPTis provided. d²×d² linearly independent input states ρ_(j) are used, andeach input state ρ_(j) is transferred to a quantum noise process toobtain an output state ε(ρ_(j)). Because of completeness of the inputstates, the output state may be represented as a linear combination ofthe input states:

ε(ρ_(j))=Σ_(k) c _(jk)ρ_(k), where c _(jk)∈

.

where ε(ρ_(j)) is an output quantum state obtained after a dynamical mapof the quantum state ρ_(j). In this way, by inputting the same quantumstate ρ_(j) a plurality of times and performing quantum state tomographyon an output state, a summation coefficient C_(jk) may be experimentallysolved. A specific process is as follows:

E _(m)ρ_(j) E _(n) ^(†)=Σ_(k) B _(m,n,j,k)ρ_(k),

where B_(m,n,j,k) is a complex number, B_(m,n,j,k) is considered as acomplex matrix formed by indexes {m, n} and {j, k}, and each of m, n, j,and k is a positive integer. Then,

Σ_(k) c _(jk)ρ_(k)=ε(ρ_(j))=Σ_(m,n,k)χ_(m,n) B _(m,n,j,k)ρ_(k).

Because {ρ_(i)} is linearly independent, the following may be obtained:

c _(jk)=Σ_(m,n)χ_(m,n) B _(m,n,j,k).

By transposing B_(m,n,j,k) , the following may be obtained:

χ_(m,n)=Σ_(j,k) B _(m,n,j,k) ⁻¹ c _(jk).

χ_(m,n) includes all information about dynamical maps of the quantumnoise process. Therefore, once χ_(m,n) is obtained through QPT, all theinformation about the dynamical maps of the quantum noise process isobtained.

However, the limited information about the quantum noise processobtained through pure QPT is insufficient to accurately andcomprehensively analyze the quantum noise process. For example, whetherthe quantum noise process is a Markov process or a non-Markov process isnot determined, a frequency spectrum of the quantum noise process is notobtained, and a correlated noise between different quantum devices inthe quantum system is not analyzed.

To resolve the foregoing technical problems, an embodiment of thisdisclosure provides a quantum noise process analysis method. FIG. 1 isan overall flowchart of a technical solution of this disclosure. In thetechnical solution provided in this disclosure, QPT is performed on aquantum noise process, to obtain dynamical maps of the quantum noiseprocess, a TTM of the quantum noise process is further extracted fromthe dynamical maps of the quantum noise process, and then the quantumnoise process is analyzed according to the TTM. The TTM is used forrepresenting a dynamical evolution of the quantum noise process, thatis, reflecting the law of evolution of the dynamical maps of the quantumnoise process over time. Therefore, richer and more comprehensiveinformation about the quantum noise process can be obtained by analyzingthe quantum noise process based on the TTM of the quantum noise processthan by pure QPT, thereby achieving a more accurate and morecomprehensive analysis of the quantum noise process.

The technical solution provided in this disclosure is applicable toanalysis of a quantum noise process of any quantum system such as aquantum computer, secure quantum communication, the quantum Internet oranother quantum system. The interference to the quantum system byquantum noise severely affects the performance of the quantum system,which is the primary barrier hindering the practical application of thequantum system. Therefore, analyzing the quantum noise process andunderstanding the properties of the noise are crucial for thedevelopment of practical quantum systems. In the technical solutionprovided in this disclosure, analyzing the quantum noise process basedon the TTM of the quantum noise process may include, for example, thefollowing analysis content as shown in FIG. 1: (1) Markov processdetermination, e.g., whether the quantum noise process is a Markovprocess or a non-Markov process can be determined, and whether a specialnoise suppression solution may be designed for a non-Markov noise, wherethe solution is, for example, suppressing the occurrence of noisethrough dynamical decoupling; (2) state evolution prediction, e.g., astate evolution of the quantum noise process may be predicted; (3)extraction of correlation function and frequency spectrum, e.g., acorrelation function and a frequency spectrum of the quantum noiseprocess may be obtained, facilitating the integration of a filter of acorresponding frequency band in the process of quantum devicemanufacturing; and (4) correlated noise analysis, e.g., a correlatednoise between different quantum devices in the quantum system may beanalyzed, to learn the source of the correlated noise and accordinglydesign a corresponding solution to suppress the correlated noise.Therefore, the technical solution provided in this disclosure can obtainricher and more comprehensive information about the quantum noiseprocess, thereby providing more information to support the improvementin the performance of the quantum system.

FIG. 2 is a flowchart of an example quantum noise process analysismethod according to an embodiment of this disclosure. The method isapplicable to a computer device, and the computer device may be anyelectronic device having data processing and storage capabilities, suchas a personal computer (PC), a server, or a computing host. The methodmay include the following steps (step 201 to step 203):

Step 201. Perform quantum process tomography (QPT) on a quantum noiseprocess of a target quantum system, to obtain dynamical maps of thequantum noise process.

The performing quantum process tomography (QPT) on a quantum noiseprocess to obtain dynamical maps of the quantum noise process has beendescribed above, so the details are not described herein again.

Optionally, in this embodiment, QPT may be performed on the quantumnoise process at discrete time points. For example, if QPT is performedat K different time points, dynamical maps of the quantum noise processat the K time points may be obtained, K being an integer greater than orequal to 1. Optionally, among the K time points, intervals betweenneighboring time points may be equal. Alternatively, intervals betweenneighboring time points may also be not equal, which is not limited inthis embodiment.

Step 202. Extract TTMs of the quantum noise process from the dynamicalmaps.

In this embodiment of this disclosure, the TTM of the quantum noiseprocess is used for representing a dynamical evolution of the quantumnoise process to reflect the law of evolution of the dynamical maps ofthe quantum noise process over time.

Optionally, if dynamical maps of the quantum noise process at the K timepoints are obtained in step 201, a possible implementation of step 201may include calculating the TTMs of the quantum noise process at the Ktime points according to the dynamical maps of the quantum noise processat the K time points. In an exemplary embodiment, the TTMs at the K timepoints are extracted recursively. For example, a TTM T_(n) of thequantum noise process at an n^(th) time point is calculated according tothe following formula:

${T_{n} \equiv {ɛ_{n} - {\sum\limits_{m = 1}^{n - 1}{T_{n - m}ɛ_{m}}}}},$

where T₁=ε₁, ε_(n) represents a dynamical map of the quantum noiseprocess at the n^(th) time point, ε_(m) represents a dynamical map ofthe quantum noise process at an m^(th) time point, and T_(n−m)represents a TTM of the quantum noise process at an (n−m)^(th) timepoint, both n and m being positive integers.

Step 203. Analyze the quantum noise process according to the TTMs.

After the TTMs of the quantum noise process at the K time points areextracted, the quantum noise process may be analyzed accordingly.

In an exemplary embodiment, after T_(n) is determined, the quantum noiseprocess may be considered as a Markov process if the value of |T_(n)| isnegligibly small for n>1 according to the definition. Otherwise, thequantum noise process may be considered as a non-Markov process. Thatis, it may be determined that the quantum noise process is a Markovprocess when each of moduli of TTMs of the quantum noise process atfirst time points is less than a preset threshold, the first time pointsbeing time points other than the foremost time point of the K timepoints. It may otherwise be determined that the quantum noise process isa non-Markov process when a modulus of a TTM of the quantum noiseprocess at a second time point is greater than the preset threshold, thesecond time point being at least one time point other than the foremosttime point of the K time points.

By means of the above method, based on the TTMs of the quantum noiseprocess, whether the quantum noise process is a Markov process or anon-Markov process can be determined, and a special noise suppressionsolution may be designed for a non-Markov noise, where the solutioninclude, for example, suppressing the occurrence of noise throughdynamical decoupling.

Additionally, compared with the dynamical map, a universal equation fordescribing the evolution of the quantum system in an open bath is anon-temporal localized quantum master equation, and can better revealthe mathematical structure of the quantum noise process. This equationis a differential-integral equation:

${\frac{d{\rho (t)}}{dt} = {{L_{s}{\rho (t)}} + {\int_{0}^{t}{{ds}\; {\kappa \left( {t - s} \right)}{\rho (s)}}}}},$

where ρ(t) represents a quantum state of the quantum system at time tand is represented by using a d×d complex matrix; L_(s) is a Liouvilleoperator and represents a coherent part in the evolution process of thequantum system; s is an integral parameter; and κ(t) is a memory kernelincluding all information about system decoherence triggered by thebath. If L_(s) and κ(t) of the quantum noise process are obtained, thenoise mechanism may be completely understood. The basic idea of thetechnical solution of this disclosure is to calculate a TTM through anexperiment and QPT, thereby extracting information about L_(s) and κ(t).

In addition, a joint evolution of the quantum system and the bath isdetermined by a joint Hamiltonian. The joint Hamiltonian may berepresented as:

$\begin{matrix}{{H(t)} = {H_{s} + {H_{sb}(t)}}} \\{{= {H_{s} + {\sum\limits_{i,\alpha}{g_{i}{B_{i}^{\alpha}(t)}\sigma_{i}^{\alpha}}}}},}\end{matrix}$

where H_(s) is a Hamiltonian of the quantum system; H_(sb) is aninteractive Hamiltonian of coupling between the quantum system and thebath; σ_(i) ^(α) is an α^(th) type of Pauli operator acting on an i^(th)qubit of the system, both i and α being positive integer indexes; B_(i)^(α)(t) is an α^(th) type of bath operator coupled to the i^(th) qubit;α=x, y, z represents three temporal-spatial directions; and g_(i) is acoupling strength between the system and the bath.

The evolution of a state function of the quantum system follows:

$\begin{matrix}{{{\rho (t)} = {{Tr}_{B}\left\lbrack {{\exp_{+}\left( {{- i}{\int_{0}^{t}{ds{H(s)}}}} \right)}{{\rho (0)} \otimes \rho_{B}}{\exp_{-}\left( {i{\int_{0}^{t}{ds{H(s)}}}} \right)}} \right\rbrack}},} \\{= {{ɛ(t)}{\rho (0)}}}\end{matrix}$

where ρ(t) represents a quantum state of the quantum system at time t,ρ(0) represents an initial quantum state of the quantum system, ρ_(B) isa quantum state of the bath, Tr_(B) represents calculation of a partialtrace of the degree of freedom of the bath, exp₊, exp⁻ are clockwise andcounterclockwise time-ordered exponential operators respectively, ε(t)represents a dynamical evolution of the quantum system at the time t, iis a unit pure imaginary number, and s is an integral parameter.

If time is discretized by t_(k+1)−t_(k)=δt (k is a positive integer), agroup of dynamical maps {ε_(k)≡ε(t_(k))} evolving over time may bedefined. Experimentally, the dynamical maps may be obtained byperforming QPT at different time points.

With reference to the foregoing definition about the formula of the TTM,by using T_(n) to express ε_(n) and substituting the expression into theformula of the foregoing state function, the following may be obtained:

${{\rho \left( t_{n} \right)} = {\sum\limits_{m = 1}^{n - 1}{T_{m}{\rho \left( t_{n - m} \right)}}}},$

where ρ(t _(n)) represents a quantum state at an n^(th) time pointt_(n), ρ(t_(n−m)) represents a quantum state at an (n−m)^(th) time pointt_(n−m), and T_(m) represents a TTM at an m^(th) time point. Thisformula clearly indicates that in the presence of a noise, the stateevolution of the quantum system depends on the historical evolution thequantum system. Generally, the dependence of the dynamical evolution onhistory does not exceed a certain time span. This means that theinfluence of the noise on the state may be precisely estimated bytruncating a convolution of the foregoing formula and keeping K (where Kis a positive integer) time points, that is, all items for which t>t_(K)are discarded. In this way, through QPT on a dynamical map in a shortperiod of time, a TTM in this period of time may be obtained. Then, anevolution of an open system in a long time may be predicted by using theTTM in this short period of time. The quantum state ρ(t_(n)) at then^(th) time point t_(n) may be calculated through the foregoing formula.In addition, the predicted quantum state may be directly compared withan experiment to verify the effectiveness of dynamics of the open systemdescribed through the TTM. In other words, this provides a preliminarybasis for determining the effectiveness of the technical solution ofthis disclosure.

To sum up, in the technical solutions provided in this disclosure, QPTis performed on a quantum noise process, to obtain dynamical maps of thequantum noise process, and a TTM of the quantum noise process is furtherextracted from the dynamical maps of the quantum noise process. The TTMis used for representing a dynamical evolution of the quantum noiseprocess to reflect the law of evolution of the dynamical maps of thequantum noise process over time. Compared with pure QPT, this disclosurecan obtain richer and more comprehensive information about the quantumnoise process. Therefore, when the quantum noise process is analyzedbased on the TTM of the quantum noise process, a more accurate andcomprehensive analysis of the quantum noise process can be achievedbased on the richer and more comprehensive information.

Additionally, in the technical solution provided in this disclosure, thedetermination of whether the quantum noise process is a Markov processor a non-Markov process according to the TTM of the quantum noiseprocess is further implemented; and the prediction, according to a TTMof the quantum noise process within a period of time, of a stateevolution of the quantum noise process within a subsequent time isfurther implemented.

In an exemplary embodiment, after the TTM of the quantum noise processis extracted, a correlation function and a frequency spectrum of thequantum noise process may be further obtained accordingly. The processmay include the following steps.

1. Extract a second-order memory kernel of the quantum noise processaccording to the TTMs of the quantum noise process when the quantumnoise process is a steady noise.

For a steady noise (for example, Gaussian steady noise), the propertiesof the noise are determined by a correlation function of the noiseprocess. The correlation function of the noise process may be calculatedaccording to a second-order memory kernel of the noise process.

In this embodiment of this disclosure, for a quantum noise process, asecond-order memory kernel of the quantum noise process is extractedaccording to the TTMs of the quantum noise process when the quantumnoise process is a steady noise.

Considering that the time has been discretized and approximated to thesecond order of a time step δt, an approximation of the TTM may beobtained:

T _(n)=(1+L _(s) δt)δ_(n,1)+κ(t _(n))δt ²,

where δt is the time step, δ_(n,1) is a Kronecker function having avalue of 1 when n=1 and a value of 0 in other cases, n being a positiveinteger; and κ(t_(n)) is a value of the memory kernel at time t_(n).

Moreover, according to the open system theory, a precise expression of adynamical memory kernel κ is:

κ(t,t′)=PL(t)exp₊[∫_(t′) ^(t) dsQL(s)]QL(t′)P,

where P=Tr_(B){ρ_(SB)}⊗ρ_(B) is a map operator, and Q_(ρ) _(SB)=ρ_(SB)−P_(ρ) _(SB) ; L is a joint Liouville operator acting on thesystem and the bath, ρ_(SB) is a joint state of the system and the bath,and Q=I−P is a difference between P and an identity operator I.

Because a noise is preliminarily controlled through engineering in anordinary quantum system, the coupling strength between the quantumsystem and the bath is relatively weak. When a target quantum system andthe bath are in a weak coupling relationship, a second-orderperturbation approximation may be established, and therefore thefollowing may be obtained:

κ₂(t)(⋅)≈<L _(sb)(t)L _(sb)(0)>(⋅)=Σ_(αα′)[σ^(α) , C _(αα′)(t)σ^(′′()t)(⋅)−C* _(αα′)(t)(⋅)σ^(α′)(t)],

where κ₂(t) is a value of a second-order memory kernel at the time t,and C*_(αα′)(t) is a complex conjugate of C_(αα′)(t). The foregoingexpression is under the Schrödinger representation, and at the sametime, it is assumed that a Hamiltonian of a joint system istime-invariant. The second-order correlation function C_(αα′)(t) isdefined as:

C _(αα′)(t)=g ²

{circumflex over (B)} ^(α)(t){circumflex over (B)} ^(α′)(0)

.

C_(αα′)(t) is a bath correlation function. Based on the second-orderperturbation, a dynamical map may be extracted from an experiment, and aTTM is obtained through QPT, thereby obtaining a memory kernel κ_(exp)through approximation. That is, κ_(exp) may be an approximatesecond-order memory kernel obtained through an experiment.

When the target quantum system and the bath are in a strong couplingrelationship, the second-order perturbation is no longer a desirableapproximation, and a better approximation can be obtained only whenthere are more high-order items, but a second-order memory kernel canstill be extracted by extracting a TTM from experimental data. Specificsteps are as follows: selecting N different parameters, performing anexperiment on the quantum noise process, and extracting memory kernelsrespectively corresponding to the N different parameters from theexperiment; and performing calculation according to the memory kernelsrespectively corresponding to the N different parameters, to obtain thesecond-order memory kernel of the quantum noise process.

First, an N-order truncated approximate memory kernel is defined:

κ(N,t)=Σ_(n=1) ^(N)κ_(2n)(t),

It is assumed that a memory kernel of a system can be approximated bythis approximate memory kernel. Then it is assumed that there is aparameter-adjustable system, parameters of the system may be adjusted ineach experiment, and a Hamiltonian of the system is:

H_(s)=ω_(s,i)σ^(z),

where i represents an experiment performed by selecting an i^(th)parameter, and σ^(z) is a Pauli z operator (₀ ¹ ⁻¹ ⁰). For the selectedparameter ω_(s,i) (where ω_(s,i) belongs to an interval that theexperiment can reach), the Hamiltonian is normalized, and the followingmay be obtained:

{tilde over (H)}=σ ^(z) +g{tilde over (H)} _(sb)(t)

where {tilde over (H)} is the normalized Hamiltonian, and g∝1/ω_(s,i).By performing an experiment on N different ω_(s,i), a group of memorykernels without physical units may be constructed, with the relationshipbeing as follows:

{tilde over (κ)}_(2n,i)(t)=γ_(i) ^(2n){tilde over (κ)}_(2n,0)(t),

where γ_(i)=ω_(s,0)/ω_(s,i) is a normalized parameter, and {tilde over(κ)}_(2n,i) represents a 2n-order memory kernel in the case of ω_(s,i).In this way, the following matrix may be defined:

$A = {\begin{bmatrix}1 & 1 & \ldots & 1 \\\gamma_{1}^{2} & \gamma_{1}^{2} & \ldots & \gamma_{1}^{2N} \\\vdots & \vdots & \ldots & \vdots \\\gamma_{{N­}\; 1}^{2} & \gamma_{{N­}\; 1}^{4} & \text{...} & \gamma_{N - 1}^{2N}\end{bmatrix}.}$

In addition, the following equation is satisfied:

A[{tilde over (κ)}_(2,0) . . . {tilde over (κ)}_(2N,0)]^(T)=[{tilde over(κ)}(1) . . . {tilde over (κ)}(N)]^(T),

where A is an N-order normalized parameter matrix, and a memory kernelon the right side of the equation may be directly extracted from anexperiment through QPT and data processing. Because A is a full-rankmatrix, a second-order memory kernel is naturally obtained by solvingthe linear equation to obtain 2-order to N-order memory kernels withoutphysical units.

2. Calculate a correlation function of the quantum noise processaccording to the second-order memory kernel of the quantum noiseprocess.

Optionally, the correlation function C_(αα′) of the quantum noiseprocess may be numerically extracted according to the following formula:

${\underset{C_{{\alpha\alpha}^{\prime}}{(t_{n})}}{argmin}\left\{ {{{{\kappa_{2}\left( {t_{n};{C_{{\alpha\alpha}^{\prime}}\left( t_{n} \right)}} \right)} - {\kappa_{\exp}\left( t_{n} \right)}}} + {\left( {1 - \delta_{t_{n},t_{0}}} \right)\lambda_{n}{\sum\limits_{\alpha,\alpha^{\prime}}{{{C_{{\alpha\alpha}^{\prime}}\left( t_{n} \right)} - {C_{{\alpha\alpha}^{\prime}}\left( t_{n - 1} \right)}}}}}} \right\}},$

where κ₂ represents the second-order memory kernel of the quantum noiseprocess, t_(n) represents the n^(th) time point, C_(αα′)(t_(n)) is asecond-order correlation function at the n^(th) time point t_(n),κ_(exp) represents an approximate second-order memory kernel obtainedthrough an experiment, δ_(t) _(n) _(,t) ₀ is a Kronecker function(having a value of 1 when n=0 and a value of 0 in other cases), λ_(n) isan adjustable parameter, and C_(αα′)(t_(n−1)) is a second-ordercorrelation function at an (n−1)^(th) time point t_(n−1). λ_(n) is usedfor ensuring that after the target function is minimized, thecorrelation function can still be continuous. λ_(n) may be determined byfirst selecting an initial value and observing the value of the targetfunction followed by iterative adjustments, so as to render theselection of the value of λ_(n) robust.

Optionally, for a non-Gaussian steady noise, only a correlation functionhigher than 2-order can fully represent statistical properties of thenoise. Obtaining of a second-order correlation function of a noise bysolving this linear equation A[{tilde over (κ)}_(2,0) . . . {tilde over(κ)}_(2N,0)]^(T)=[{tilde over (κ)}(1) . . . {tilde over (κ)}(N)]^(T) hasbeen described above. If a non-Gaussian steady noise is processed, itmay be assumed that a memory kernel of the noise is written as:

κ(N,t)=Σ_(n=2) ^(N)κ_(n)(t) .

Based on this more generalized memory kernel, the following may beobtained according to the solution described above:

A[{tilde over (κ)}_(2,0) . . . {tilde over (κ)}_(N,0)]^(T)=[{tilde over(κ)}(1) . . . {tilde over (κ)}(N)]^(T).

By solving this linear equation, second-order and higher-ordercorrelation functions may be obtained.

3. Perform a Fourier transform on the correlation function of thequantum noise process, to obtain a frequency spectrum of the quantumnoise process.

Once the correlation function of the quantum noise process is obtained,a Fourier transform may be performed on the correlation function, toobtain a frequency spectrum J_(αα′)(ω) of the quantum noise process:

J _(αα′)(ω)=1/2∫_(−∞) ^(∞) dte ^(iωt)[C _(αα′)(t)−C* _(αα′)(t)].

This method of obtaining the frequency spectrum of the quantum noiseprocess is not limited by whether the noise is a quantum noise (thesystem has a feedback to the noise source) or a classical noise, and isnot limited by a particular noise type.

To sum up, in the technical solution provided in this disclosure, afterthe TTM of the quantum noise process is extracted, the correlationfunction and the frequency spectrum of the quantum noise process may befurther obtained accordingly, facilitating the integration of a filterof a corresponding frequency band in the process of quantum devicemanufacturing.

In an exemplary embodiment, after the TTM of the quantum noise processis extracted, a correlated noise between different quantum devices inthe target quantum system may be further analyzed accordingly to learnthe source of the correlated noise. The process may include thefollowing steps.

1. Calculate, for s quantum devices included in the target quantumsystem, a correlated TTM among the s quantum devices according to TTMsrespectively corresponding to the s quantum devices, s being an integergreater than 1.

2. Analyze a source of a correlated noise among the s quantum devicesaccording to the correlated TTM.

A quantum system may include a plurality of quantum devices. A qubit isthe simplest quantum device, including only two quantum states. By usingTTMs, a noise correlation between the plurality of quantum devices inthe same quantum system can be completed. The following mainly describesan example scenario where there are two quantum devices, and the sameapplies to other scenarios. For example, according to the methodprovided in this embodiment of this disclosure, a noise correlationbetween any two quantum devices, or a noise correlation between anythree or more quantum devices can also be determined.

Dynamical maps of any two quantum systems (or quantum devices) may bedecomposed as follows:

$\begin{matrix}{ɛ_{n} \equiv {{ɛ_{n,1} \otimes ɛ_{n,2}} + {\delta ɛ}_{n}}} \\{= {{\overset{\_}{ɛ}}_{n} + {\delta ɛ}_{\;_{n}}}}\end{matrix},$

where ε_(n,1) represents a dynamical map of a first quantum device,ε_(n,2) represents a dynamical map of a second quantum device, andδε_(n) is an unseparated part representing the influence of a correlatednoise. In the foregoing decomposition of dynamical maps, a dynamical mapε_(n)→χ_(n) may be expressed in the form of Choi matrix, that is, χ_(n)is a Choi matrix being an equivalent representation of the dynamicalmap, and a trace of the Choi matrix is calculated as follows:

Tr _(īχn)=χ_(n,i), (i,ī)=(1, 2), or (i, ī)=(2, 1).

Then, a Choi matrix χ_(n,i) of a single quantum device is expressed backas a dynamical map ε_(n,i). Dynamical maps ε_(n) of two quantum devicesmay both be obtained by performing joint QPT on the two quantum devices.δε_(n) may be used for analyzing the correlated noise. In the case ofsecond-order perturbation, a modulus of δε_(n) is usually much less thanthat of ε _(n). In a non-perturbative area, modulus values of δε_(n) andε _(n) may be equivalent, or even |δε_(n)| is much greater thanTherefore, pure QPT can provide a preliminary determination on thestrength of the correlated noise. However, it is rather difficultanalyze the source of the correlated noise because all data is mixedtogether. Usually, sources of the correlated noise between two quantumdevices include: (1) a correlated noise generated from direct couplingbetween the two quantum devices; (2) a correlated noise induced by ashared bath of the two quantum devices; or (3) a combination thereof.

An embodiment of this disclosure provides a correlated noise analysismethod based on a TTM. By this method, more information about acorrelated noise may be obtained. First, a separable TTM is calculatedaccording to ε _(n):

${{\overset{\_}{T}}_{n} = {{\overset{\_}{ɛ}}_{n} - {\sum\limits_{m = 1}^{n - 1}{{\overset{\_}{T}}_{n - m}{\overset{\_}{ɛ}}_{m}}}}},$

and

T _(n) =T _(n) +δT _(n),

where T ₁=ε ₁, and δT_(n) is a noise correlation in transfer tensor map.Similarly, δT_(n) may be decomposed into:

δT _(n) =δLδtδ _(n,1)+δκ_(n) δt ²,

where the Liouville super-operator δL reveals whether there is acorrelated noise generated from direct coupling between two quantumdevices, and δκ_(n) represents a correlated noise induced by a sharedbath. It can be found that a coupling increment caused by δL is in alinear relationship with δt, and a coupling increment caused by δκ_(n)is in a linear relationship with δt². By selecting two different timesteps δt and δt′, two different dynamical maps ε₁ and ε′₁ may begenerated, and then the source of the correlated noise is determined.Considering that the correlated noise has significant influence onfault-tolerant quantum computing, the technical solution of thisdisclosure can provide a better understanding of the correlated noiseand provide guidance on how to perform control, so as to designdifferent noise suppression solutions.

For example, through QPT, a joint dynamical map ε_(n) of two quantumdevices in the target quantum system may be obtained, a TTM T_(n) isfurther obtained, then ε_(n,1) and ε_(n,2) may be obtained bycalculating traces for the two quantum devices according to ε_(n)respectively, and ε _(n)=ε_(n,1)⊗ε_(n,2) . Then, T _(n) is obtainedthrough

${\overset{\_}{T}}_{n} = {{\overset{\_}{ɛ}}_{n} - {\sum\limits_{m = 1}^{n - 1}{{\overset{\_}{T}}_{n - m}{{\overset{\_}{ɛ}}_{m}.}}}}$

Finally, δT_(n)=T_(n)−T _(n)=δLδtδ_(n,1)+δκ_(n)δt².

Then, considering two different time steps δt and δt′:

$\left\{ {\begin{matrix}{{\delta T_{n}} = {{\delta L\delta t\delta_{n,1}} + {\delta \kappa_{n}\delta t^{2}}}} \\{{\delta \; {T^{\prime}}_{n}} = {{\delta \; L\; \delta \; t^{\prime}\; \delta_{n,1}} + {\delta \kappa_{n}{{\delta t}^{\prime}}^{2}}}}\end{matrix},} \right.$

δL and δκ_(n) are calculated respectively.

To sum up, in the technical solution provided in this disclosure, afterthe TTM of the quantum noise process is extracted, a correlated noisebetween different quantum devices in the target quantum system may befurther analyzed accordingly, to learn the source of the correlatednoise and design a corresponding solution to suppress the correlatednoise.

To further verify the effectiveness of the technical solution of thisdisclosure, numerical analysis is performed for a typical model. Afterthis, on IBM Quantum Experience (which is a quantum computing cloudplatform provided by IBM), an attempt is made to perform an experimentand observation on a real superconducting qubit and extract informationabout a noise process from the real superconducting qubit by using thetechnical solution of this disclosure.

Results of numerical analysis for the typical model are as follows:

Case 1: numerical simulation of a single qubit under pure phasedecoherence

Letting H_(s)=0.1σ_(z), H_(sb)=B^(z)((t)σ^(z),C_(zz)(0)=λ=4,δt=0.2, TTMresults of a free evolution of a single qubit are as shown in FIG. 3.Part (a) in FIG. 3 shows the variation of a Frobenius norm of the TTMover time. It may be seen from the figure that the TTM within a ranget₁→t₆ makes a non-trivial contribution, that is, non-Markov propertiesare demonstrated in the noise process. A line 31 in part (b) in FIG. 3represents an evolution of a real part of a non-diagonal element of adensity matrix corresponding to an initial state (|0

+|1

)/√{square root over (2)} over time. A line 32, a line 33 and a line 34respectively present prediction effects for the density matrix atdifferent TTM lengths (that is, when K is 1, 3, and 5 respectively). Itmay be seen that when K is 5, an evolution obtained through the TTM wellcoincides with an exact solution, and a long-term experimental evolutioncan be perfectly predicted.

Additionally, FIG. 4 shows the variation of a Bloch volume over time,and the increase in the Bloch volume V(t) within a period of time (t₄,t₅, t₆)demonstrates non-Markov properties of the dynamical process,which proves the conclusion in FIG. 3 from another perspective.

Case 2: extraction of a noise correlation function of a single qubitunder pure phase decoherence

Letting H_(s)=0.02σ_(z),C_(zz) (0)=

B^(z)(t)B^(z)(0)

=0.01, δt=0.04, and a frequency spectrum of a quantum noise process(equivalent to a bath noise spectrum) is obtained through a TTM methodfor a free evolution of a single qubit. Part (a) in FIG. 5 shows thevariation of a noise correlation function C^(zz) (t) over time in thecase of weak coupling between the quantum system and the bath. A line 51presents an accurate theoretical result of the noise correlationfunction, and each circle presents a numerical result obtained from amemory kernel under an assumption of K(t)≈K₂(t) . It can be learned thatin the case of weak coupling, an approximate second-order memory kernelobtained from TTM can well depict the quantum noise process.

In part (b) in FIG. 5, let H_(s)=0.02σ_(z),C_(zz)(0)∈(0, 2.56), δt=0.04.In the case of strong coupling between the quantum system and the bath,a noise correlation function at a special time point C_(zz)(t=15δt)changes with a coupling strength C_(zz)(0)λ between the noise and thesystem. Line 52 represents an accurate theoretical result. Line 53presents a first numerical result: that is, directly assumingK(t)≈K₂(t), it can be learned that in the case of strong coupling, thereis a big difference between the first numerical result and a real noisespectrum. line 54 presents a second numerical result: that is, directlyassuming K(t)≈K₂(t)+K₄(t), even in the case of strong coupling underresearch, a memory kernel obtained from TTM can well reflect the realnoise spectrum even for higher order.

Case 3: extraction of a correlation function of a single qubit in a bitflip noise

Letting H_(s)=0.02δ_(z),C_(xx)(0)=0.01, δt=0.04, a frequency spectrum ofa quantum noise process (equivalent to a bath noise spectrum) isobtained through a TTM method for a free evolution of a single qubit. Inthis case, the noise is no longer a pure phase decoherence noise. Asshown in FIG. 6, a correlation function C_(xx)(t)=

B^(x)(t)B^(x)(0)

presented by a circle which is obtained from a memory kernel of a TTMwell coincides with a real noise spectrum presented by line 61. Thisgroup of simulation indicates that when the influence of the bath noiseexceeds that of pure dephasing, for example, B^(x)(t), B^(y)(t), themethod of deducing a noise spectrum from a memory kernel of a TTM isstill applicable.

Case 4: TTM results of free evolutions of two qubits in two double-qubitpure dephasing models

Part (a) in FIG. 7 shows TTM results of free evolutions of two qubitscoupled to each other in a direction z when the two qubits are locatedin respective independent bath noises.

A system Hamiltonian is: H_(s)=ω₁σ₁ ^(z)+ω₂σ₂ ^(z)+ω₁₂σ₁ ^(z)σ₂^(z),ω₂=0.1,ω₁₂=0.05.

A bath Hamiltonian is: H=B₁ ^(z)(t)σ₁ ^(z)+B₂ ^(z)(t)σ₂ ^(z),

A correlation function is: C_(zz)(0)=

B₁ ^(z)(t)B₁ ^(z)(0)

=

B₂ ^(z)(t)B₂ ^(z)(0)

=1,

B₁ ^(z)(t)B₂ ^(z)(t′)

=0,δt=0.2.

Line 71, line 72 and line 73 represent a full TTM T_(n), a separable TTMT _(n) and a correlated TTM δT_(n), respectively. As shown in thefigure, only the first item, that is, δT₁, in the correlated TTM isnon-trivial. The result indicates that in an independent noise bath, acorrelated part of a TTM is almost Markovian. Further, it can be learnedthrough analysis that an entanglement of two qubits generated by δL_(s)leads to a correlated decoherence effect even if noise sources areseparated or independent of each other in sapce.

Part (b) in FIG. 7 shows TTM results of free evolutions of two qubitsnot directly coupled to each other when the two qubits are located incorrelated bath noises.

A system Hamiltonian is: H_(s)=ω₁σ₁ ^(z)+ω₂σ₂ ^(z)ω₁=ω₂=0.1

A bath Hamiltonian is: H_(sb)=B₁ ^(z)(t)σ₁ ^(z)+B₂ ^(z)(t)σ₂^(z),C_(zz)(0)=

B₁ ^(z)(t)B₁ ^(z)(0)

=

B₂ ^(z)(t)B₂ ^(z)(0)

=

B₁ ^(z)(t)B₂ ^(z)(t′)

=1,δt=0.2.

Line 74, line 75 and line 76 represent a full TTM T_(n), a separable TTMT _(n) and a correlated TTM δT_(n), respectively. In this case, aplurality of δT_(n) are non-trivial. It may be found through analysisthat δK(t₁) is the main contributing factor of δT₁. Therefore, relativeimportance of different physical mechanisms that cause collectivedecoherence can be estimated directly according to the norm distributionof TTMs over time.

Case 5: prediction effectiveness of dynamics of an open system ofdouble-qubit TTMs.

To investigate the importance of δT_(n), FIG. 8 presents a dynamicalevolution of non-diagonal matrix elements of a density matrix of twoqubits. Prediction results of TTMs whose lengths are (that is, K is) 1,8, and 16 in a physical state are compared with a real dynamicalsimulation result. Parts (a) and (b) in FIG. 8 respectively presentprediction results for |ψ(0)

=(|00

+|10

)/√{square root over (2)} based on a full TTM and a separable TTM in afirst model. Parts (c) and (d) in FIG. 8 respectively present predictionresults for |ψ(0)

=(|01

+|10

)/√{square root over (2)} based on a full TTM and a separable TTM in asecond model. In both the two cases, the effect of collectivedecoherence cannot be described by using T _(n) alone. In FIG. 7, δT_(n)is very small, and has little influence. However, as can be seen fromFIG. 8, δT_(n) still plays an important role in the prediction of thephysical state. This further proves the complex characteristics of ahighly non-Markovian system.

Additionally, to verify the practicability of the technical solution ofthis disclosure, tests have been conducted on IBM Quantum Experience.IBM Quantum Experience is a superconducting quantum computing cloudplatform provided by IBM, and all computations are run on a realsuperconducting quantum computer. For a superconducting qubit, becausethe time for operating a quantum gate is too long (about 100 ns)relative to the correlation time of the bath and the noise process isnot pure phase decoherence, the method of extracting a frequencyspectrum based on dynamical decoupling of CPMG is inapplicable.

FIG. 9 shows research on TTMs of a free evolution of a single qubit onan IBM quantum computing cloud platform “IBM 16 Melbourne”, where δt=2.2μs. Part (a) in FIG. 9 shows the norm distribution of TTMs over time.Part (b) in FIG. 9 shows a dynamical evolution of a state |1

, and line 91 is an experimental result. Line 92, line 93 and line 94are respective prediction results for an evolution of |1

when (1, 3, and 5) TTMs are taken respectively. It may be seen that thetime scale of the memory kernel is in the order of magnitude of μs,which is not short compared with the quantum gate time of 100 ns.

FIG. 10 shows the distribution of a Bloch volume V(t) of a single qubitover time. A transient increase demonstrates the non-Markovcharacteristics of the quantum system.

FIG. 11 shows a dynamic decoupling (DD) evolution of a single qubit onan IBM quantum computing cloud platform “IBM 16 Melbourne”, whereδt=2.64 μs. Measurement results of four initial states (a) |ψ(0)

=|0

, (b) |ψ(0)

=|1

, (c) |ψ(0)

=(|0

+|1

)/√{square root over (2)} and (d) |ψ(0)

=(|0

+i|1

)/√{square root over (2)} in three spinning directions under the XY4DDprotocol are shown. Extension of quantum coherence may be observed.

FIG. 12 shows a dynamic decoupling (DD) evolution of a single qubit onan IBM quantum computing cloud platform “IBM 16 Melbourne”, where δt=2.64 μs. The norm distribution of TTMs over time under the XY4DD protocolis shown. An internal mechanism of extension of quantum coherence may bereflected by this TTM: an effective noise under the XY4DD protocol ismore Markovian than a result of a free evolution.

FIG. 13 shows research on a TTM of a free evolution of two qubits on anIBM quantum computing cloud platform “IBM 16 Melbourne”, where δt=2. 2μs. The norm distribution of a full TTM |T_(n)|, a separable TTM |T_(n)|, and a correlated TTM |δT_(n)| over time is shown. It may be seenthat in this group of experiments, the TTMs are all non-trivial in arelatively long time scale, and have relatively strong non-Markovproperties. With reference to the result of numerical simulation, it maybe preliminarily considered that there are inter-bit coupling and a bathnoise correlation between two neighboring bits on the IBM quantum cloudplatform.

FIG. 14 shows research on TTMs of free evolutions of two qubits on anIBM quantum computing cloud platform “IBM 16 Melbourne”, where δt=2.2μs. The black line with black dots is an experimental result of anevolution of a density matrix, and three lines represented with circles,triangles and squares are respectively results of predicting anevolution of a density matrix by selecting (1, 2, and 4) TTMsrespectively. Parts (a) and (b) in FIG. 14 respectively presentprediction results based on a full TTM and a separable TTM when theinitial state is a non-entangled state |ψ(0)

=(|11

+|10

)/√{square root over (2)}. Parts (c) and (d) in FIG. 14 respectivelypresent prediction results based on a full TTM and a separable TTM whenan initial state is an entangled state |ψ(0)

=(|00

+|11

)/√{square root over (2)}. It can be learned that if no correlated TTMis included, the evolution cannot be accurately predicted either in theentangled state or in the non-entangled state. Through a furtheranalysis of δT₁, it may be seen that δL_(s) makes an importantcontribution, indicating that the two qubits are coupled to each other.

The following is an apparatus embodiment of this disclosure, which canbe used to execute the method embodiments of this disclosure. Fordetails that are not disclosed in the apparatus embodiments of thisdisclosure, refer to the method embodiments of this disclosure.

FIG. 15 is a block diagram of a quantum noise process analysis apparatusaccording to an embodiment of this disclosure. The apparatus hasfunctions of implementing the foregoing method embodiments. Thefunctions may be implemented by hardware, or may be implemented byhardware executing corresponding software. The apparatus may be acomputer device, or may be disposed in a computer device. The apparatus1500 may include: an obtaining module 1510, an extraction module 1520,and an analysis module 1530.

The obtaining module 1510 is configured to perform quantum processtomography (QPT) on a quantum noise process of a target quantum system,to obtain dynamical maps of the quantum noise process.

The extraction module 1520 is configured to extract TTMs of the quantumnoise process from the dynamical maps, the TTMs being used forrepresenting a dynamical evolution of the quantum noise process.

The analysis module 1530 is configured to analyze the quantum noiseprocess according to the TTMs.

To sum up, in the technical solutions provided in this disclosure, QPTis performed on a quantum noise process, to obtain dynamical maps of thequantum noise process, and a TTM of the quantum noise process is furtherextracted from the dynamical maps of the quantum noise process. The TTMis used for representing a dynamical evolution of the quantum noiseprocess, that is, reflecting the law of evolution of the dynamical mapsof the quantum noise process over time. Compared with pure QPT, thisdisclosure can obtain richer and more comprehensive information aboutthe quantum noise process. Therefore, when the quantum noise process isanalyzed based on the TTM of the quantum noise process, a more accurateand comprehensive analysis of the quantum noise process can be achievedbased on the richer and more comprehensive information.

In some possible designs, the dynamical maps include dynamical maps ofthe quantum noise process at K time points, K being a positive integer;and the extraction module 1520 is configured to calculate TTMs of thequantum noise process at the K time points according to the dynamicalmaps of the quantum noise process at the K time points.

In some possible designs, the extraction module 1520 is configured tocalculate a TTM T_(n) of the quantum noise process at an n^(th) timepoint according to the following formula:

${T_{n} \equiv {ɛ_{n} - {\sum\limits_{m = 1}^{n - 1}{T_{n - m}ɛ_{m}}}}},$

where T₁=ε₁, ε_(n) represents a dynamical map of the quantum noiseprocess at the n^(th) time point, ε_(m) represents a dynamical map ofthe quantum noise process at an m^(th) time point, and T_(n−m)represents a TTM of the quantum noise process at an (n−m)^(th) timepoint, both n and m being positive integers.

In some possible designs, as shown in FIG. 16, the analysis module 1530includes a Markov determination sub-module 1531.

The Markov determination sub-module 1531 is configured to:

determine that the quantum noise process is a Markov process when eachof moduli of TTMs of the quantum noise process at first time points isless than a preset threshold, the first time points being time pointsother than the foremost time point of the K time points; and

determine that the quantum noise process is a non-Markov process when amodulus of a TTM of the quantum noise process at a second time point isgreater than the preset threshold, the second time point being at leastone time point other than the foremost time point of the K time points.

In some possible designs, as shown in FIG. 16, the analysis module 1530includes a state evolution prediction sub-module 1532.

The state evolution prediction sub-module 1532 is configured to predicta state evolution of the quantum noise process within a subsequent timeaccording to the TTMs at the K time points.

In some possible designs, the state evolution prediction sub-module 1532is configured to calculate a quantum state ρ(t_(n)) of the quantum noiseprocess at an n^(th) time point t_(n) according to the followingformula:

${{\rho \left( t_{n} \right)} = {\sum\limits_{m = 1}^{n - 1}{T_{m}{\rho \left( t_{n - m} \right)}}}},$

where T_(m) represents a TTM at an m^(th) time point, and ρ(t_(n−m))represents a quantum state at an (n−m)^(th) time point t_(n−m), both nand m being positive integers.

In some possible designs, as shown in FIG. 16, the analysis module 1530includes:

a memory kernel extraction sub-module 1533, configured to extract asecond-order memory kernel of the quantum noise process according to theTTMs when the quantum noise process is a steady noise;

a correlation function calculation sub-module 1534, configured tocalculate a correlation function of the quantum noise process accordingto the second-order memory kernel of the quantum noise process; and

a frequency spectrum obtaining sub-module 1535, configured to perform aFourier transform on the correlation function of the quantum noiseprocess, to obtain a frequency spectrum of the quantum noise process.

In some possible designs, the memory kernel extraction sub-module 1533is configured to: select N different parameters, perform an experimenton the quantum noise process, and extract memory kernels respectivelycorresponding to the N different parameters from the experiment; andperform calculation according to the memory kernels respectivelycorresponding to the N different parameters, to obtain the second-ordermemory kernel of the quantum noise process.

In some possible designs, the correlation function calculationsub-module 1534 is configured to numerically extract the correlationfunction of the quantum noise process according to the followingformula:

${\underset{C_{{\alpha\alpha}^{\prime}}{(t_{n})}}{argmin}\left\{ {{{{\kappa_{2}\left( {t_{n};{C_{{\alpha\alpha}^{\prime}}\left( t_{n} \right)}} \right)} - {\kappa_{\exp}\left( t_{n} \right)}}} + {\left( {1 - \delta_{t_{n},t_{0}}} \right)\lambda_{n}{\sum\limits_{\alpha,\alpha^{\prime}}{{{C_{{\alpha\alpha}^{\prime}}\left( t_{n} \right)} - {C_{{\alpha\alpha}^{\prime}}\left( t_{n - 1} \right)}}}}}} \right\}},$

where κ₂ represents the second-order memory kernel of the quantum noiseprocess, t_(n) represents the n^(th) time point, κ_(exp) is asecond-order correlation function at the n^(th) time point, κ_(exp)represents an approximate second-order memory kernel obtained through anexperiment, δ_(t) _(n) _(,t) ₀ represents an interval between the n^(th)time point and an initial moment, λ_(n) is an adjustable parameter, andC_(αα′)(t_(n−1)) is a second-order correlation function at an (n−1)^(th)time point t_(n−1).

In some possible designs, as shown in FIG. 16, the analysis module 1530includes a correlated noise analysis sub-module 1536.

The correlated noise analysis sub-module 1536 is configured tocalculate, for s quantum devices included in the target quantum system,a correlated TTM among the s quantum devices according to TTMsrespectively corresponding to the s quantum devices, s being an integergreater than 1; and analyze a source of a correlated noise among the squantum devices according to the correlated TTM.

When the apparatus provided in the foregoing embodiments implements itsfunctions, a description is given only by using the foregoing divisionof function modules as an example. In actual applications, the functionsmay be allocated to and implemented by different function modulesaccording to the requirements, that is, the internal structure of thedevice may be divided into different function modules, to implement allor some of the functions described above. In addition, the apparatus andmethod embodiments provided in the foregoing embodiments follow similarunderlying principles. For the specific implementation process, refer tothe method embodiments, so the details are not described herein again.

FIG. 17 is a schematic structural diagram of a computer device accordingto an embodiment of this disclosure. The computer device is configuredto implement the quantum noise process analysis method provided in theforegoing embodiments. Specifically:

The computer device 1700 includes a central processing unit (CPU) 1701,a system memory 1704 including a random access memory (RAM) 1702 and aread-only memory (ROM) 1703, and a system bus 1705 connecting the systemmemory 1704 and the CPU 1701. The computer device 1700 further includesa basic input/output system (I/O system) 1706 configured to transmitinformation between components in the computer, and a mass storagedevice 1707 configured to store an operating system 1713, an applicationprogram 1714, and other program module 1715.

The basic I/O system 1706 includes a display 1708 configured to displayinformation and an input device 1709 configured for a user to inputinformation, such as a mouse or a keyboard. The display 1708 and theinput device 1709 are both connected to the CPU 1701 by an input/outputcontroller 1710 connected to the system bus 1705. The basic I/O system1706 may further include the input/output controller 1710, to receiveand process inputs from multiple other devices, such as a keyboard, amouse, or an electronic stylus. Similarly, the input/output controller1710 further provides an output to a display screen, a printer, or othertype of output device.

The mass storage device 1707 is connected to the CPU 1701 by a massstorage controller (not shown) connected to the system bus 1705. Themass storage device 1707 and an associated computer-readable mediumprovide non-volatile storage for the computer device 1700. That is, themass storage device 1707 may include a computer-readable medium (notshown), such as a hard disk or a CD-ROM drive.

Without loss of generality, the computer-readable medium may include acomputer storage medium and a communication medium. The computer storagemedium includes volatile and non-volatile, removable and non-removablemedia implemented by using any method or technology for storinginformation such as computer-readable instructions, data structures,program modules, or other data. The computer storage medium includes aRAM, a ROM, an EPROM, an EEPROM, a flash memory, or other solid-statestorage technique, a CD-ROM, a DVD, or other optical storage, a magneticcassette, a magnetic tape, a magnetic disk storage, or other magneticstorage device. Certainly, it is known to a person skilled in the artthat the computer storage medium is not limited to the foregoing types.The system memory 1704 and the mass storage device 1707 may becollectively referred to as a memory.

According to the embodiments of this disclosure, the computer device1700 may further be connected, through a network such as the Internet,to a remote computer on the network. That is, the computer device 1700may be connected to a network 1712 by a network interface unit 1711connected to the system bus 1705, or may be connected to another type ofnetwork or remote computer system (not shown) by a network interfaceunit 1711.

The memory stores at least one instruction, at least one section ofprogram, a code set or an instruction set, and the at least oneinstruction, the at least one section of program, the code set or theinstruction set is configured to be executed by one or more processorsto implement the quantum noise process analysis method provided in theforegoing embodiments.

In an exemplary embodiment, a computer-readable storage medium isfurther provided, the storage medium storing at least one instruction,at least one program, a code set or an instruction set, the at least oneinstruction, the at least one program, the code set or the instructionset being executed by a processor of a computer device to implement thequantum noise process analysis method provided in the foregoingembodiments. In an exemplary embodiment, the computer-readable storagemedium may be a ROM, a RAM, a CD-ROM, a magnetic tape, a floppy disk, oran optical data storage device

In an exemplary embodiment, a computer program product is provided. Whenexecuted, the computer program product is configured to implement thequantum noise process analysis method provided in the foregoingembodiments.

It is to be understood that “plurality of” mentioned in thisspecification means two or more. “And/or” describes an associationrelationship between associated objects and means that threerelationships may exist. For example, A and/or B may represent thefollowing three cases: Only A exists, both A and B exist, and only Bexists. The character “/” in this specification generally indicates an“or” relationship between the associated objects. In addition, the stepnumbers described in this specification merely exemplarily show apossible execution order of the steps. In some other embodiments, thesteps may not be performed according to the order of the numbers. Forexample, two steps denoted by different numbers may be performedsimultaneously, or two steps denoted by different numbers may beperformed in an order that is reverse to that shown in the figure, whichis not limited in the embodiments of this disclosure.

The foregoing descriptions are merely examples of the embodiments ofthis disclosure, but are not intended to limit this disclosure. Anymodification, equivalent replacement, or improvement made withoutdeparting from the spirit and principle of this disclosure shall fallwithin the protection scope of this application.

What is claimed is:
 1. A quantum noise process analysis method,applicable to a computer device, the method comprising: performingquantum process tomography (QPT) on a quantum noise process of a targetquantum system, to obtain dynamical maps of the quantum noise processwherein the QPT involves at least one measurement of the target quantumsystem; extracting transfer tensor maps (TTMs) of the quantum noiseprocess from the dynamical maps, the TTMs being used for representing adynamical evolution of the quantum noise process; and analyzing thequantum noise process according to the TTMs.
 2. The method according toclaim 1, wherein: the dynamical maps comprise maps of the quantum noiseprocess at K time points, K being a positive integer; and extracting theTTMs of the quantum noise process from the dynamical maps comprises:calculating the TTMs of the quantum noise process at the K time pointsaccording to the maps of the quantum noise process at the K time points.3. The method according to claim 2, wherein calculating the TTMs of thequantum noise process at the K time points according to the maps of thequantum noise process at the K time points comprises: calculating a TTMT_(n) of the quantum noise process at an n^(th) time point according tothe following formula:${T_{n} \equiv {ɛ_{n} - {\sum\limits_{m = 1}^{n - 1}{T_{n - m}ɛ_{m}}}}},$wherein T₁=ε₁, ε_(n) represents a dynamical map of the quantum noiseprocess at the n^(th) time point, ε_(m) represents a dynamical map ofthe quantum noise process at an m^(th) time point, and T_(n−m)represents a TTM of the quantum noise process at an (n−m)^(th) timepoint, both n and m being positive integers.
 4. The method according toclaim 2, wherein analyzing the quantum noise process according to theTTMs comprises: determining that the quantum noise process is a Markovprocess when each of moduli of TTMs of the quantum noise process atfirst time points is less than a preset threshold, the first time pointsbeing time points other than the foremost time point of the K timepoints; and determining that the quantum noise process is a non-Markovprocess when a modulus of a TTM of the quantum noise process at a secondtime point is greater than the preset threshold, the second time pointbeing at least one time point other than the foremost time point of theK time points.
 5. The method according to claim 2, wherein analyzing thequantum noise process according to the TTMs comprises: predicting astate evolution of the quantum noise process within a subsequent timeaccording to the TTMs at the K time points.
 6. The method according toclaim 5, wherein predicting the state evolution of the quantum noiseprocess within the subsequent time according to the TTMs at the K timepoints comprises: calculating a quantum state ρ(t_(n)) of the quantumnoise process at an n^(th) time point t_(n) according to the followingformula:${{\rho \left( t_{n} \right)} = {\sum\limits_{m = 1}^{n - 1}{T_{m}{\rho \left( t_{n - m} \right)}}}},$wherein T_(m) represents a TTM at an m^(th) time point, and ρ(t_(n−m))represents a quantum state at an (n−m)^(th) time point t_(n−m), both nand m being positive integers.
 7. The method according to claim 1,wherein analyzing the quantum noise process according to the TTMscomprises: extracting a second-order memory kernel of the quantum noiseprocess according to the TTMs when the quantum noise process is a steadynoise; calculating a correlation function of the quantum noise processaccording to the second-order memory kernel of the quantum noiseprocess; and performing a Fourier transform on the correlation functionof the quantum noise process to obtain a frequency spectrum of thequantum noise process.
 8. The method according to claim 7, whereinextracting the second-order memory kernel of the quantum noise processaccording to the TTMs at the K time points comprises: selecting Ndifferent parameters, performing an experiment on the quantum noiseprocess, and extracting memory kernels respectively corresponding to theN different parameters from the experiment; and performing calculationaccording to the memory kernels respectively corresponding to the Ndifferent parameters, to obtain the second-order memory kernel of thequantum noise process.
 9. The method according to claim 7, whereincalculating the correlation function of the quantum noise processaccording to the second-order memory kernel of the quantum noise processcomprises: numerically extracting the correlation function C_(αα′) ofthe quantum noise process according to the following formula:${\underset{C_{{\alpha\alpha}^{\prime}}{(t_{n})}}{argmin}\left\{ {{{{\kappa_{2}\left( {t_{n};{C_{{\alpha\alpha}^{\prime}}\left( t_{n} \right)}} \right)} - {\kappa_{\exp}\left( t_{n} \right)}}} + {\left( {1 - \delta_{t_{n},t_{0}}} \right)\lambda_{n}{\sum\limits_{\alpha,\alpha^{\prime}}{{{C_{{\alpha\alpha}^{\prime}}\left( t_{n} \right)} - {C_{{\alpha\alpha}^{\prime}}\left( t_{n - 1} \right)}}}}}} \right\}},$wherein κ₂ represents the second-order memory kernel of the quantumnoise process, t_(n) represents the n^(th) time point, C_(αα′)(t_(n)) isa second-order correlation function at the n^(th) time point t_(n),κ_(exp) represents an approximate second-order memory kernel obtainedthrough an experiment, δ_(t) _(n) _(,t) ₀ is a Kronecker function, λ_(n)is an adjustable parameter, and C_(αα′)(t_(n−1)) is a second-ordercorrelation function at an (n−1)^(th) time point t_(n−1).
 10. The methodaccording to claim 1, wherein analyzing the quantum noise processaccording to the TTMs comprises: calculating, for s quantum devicescomprised in the target quantum system, a correlated TTM among the squantum devices according to TTMs respectively corresponding to the squantum devices, s being an integer greater than 1; and analyzing asource of a correlated noise among the s quantum devices according tothe correlated TTM.
 11. A quantum noise process analysis device,comprising a memory for storing instructions and a processor incommunication with the processor, wherein the processor, when executingthe instructions, causes the device to: perform quantum processtomography (QPT) on a quantum noise process of a target quantum system,to obtain dynamical maps of the quantum noise process wherein the QPTinvolves at least one measurement of the target quantum system; extracttransfer tensor maps (TTMs) of the quantum noise process from thedynamical maps, the TTMs being used for representing a dynamicalevolution of the quantum noise process; and analyze the quantum noiseprocess according to the TTMs.
 12. The quantum noise process analysisdevice according to claim 11, wherein: the dynamical maps comprise mapsof the quantum noise process at K time points, K being a positiveinteger; and to extract the TTMs of the quantum noise process from thedynamical maps comprises: calculate the TTMs of the quantum noiseprocess at the K time points according to the maps of the quantum noiseprocess at the K time points.
 13. The quantum noise process analysisdevice according to claim 12, wherein to calculate the TTMs of thequantum noise process at the K time points according to the maps of thequantum noise process at the K time points comprises: calculate a TTMT_(n) of the quantum noise process at an n^(th) time point according tothe following formula:${T_{n} \equiv {ɛ_{n} - {\sum\limits_{m = 1}^{n - 1}{T_{n - m}ɛ_{m}}}}},$wherein T₁=ε₁, ε_(n) represents a dynamical map of the quantum noiseprocess at the n^(th) time point, ε_(m) represents a dynamical map ofthe quantum noise process at an m^(th) time point, and T_(n−m)represents a TTM of the quantum noise process at an (n−m)^(th) timepoint, both n and m being positive integers.
 14. The quantum noiseprocess analysis device according to claim 12, wherein to analyze thequantum noise process according to the TTMs comprises: determine thatthe quantum noise process is a Markov process when each of moduli ofTTMs of the quantum noise process at first time points is less than apreset threshold, the first time points being time points other than theforemost time point of the K time points; and determine that the quantumnoise process is a non-Markov process when a modulus of a TTM of thequantum noise process at a second time point is greater than the presetthreshold, the second time point being at least one time point otherthan the foremost time point of the K time points.
 15. The quantum noiseprocess analysis device according to claim 12, wherein to analyze thequantum noise process according to the TTMs comprises: predict a stateevolution of the quantum noise process within a subsequent timeaccording to the TTMs at the K time points.
 16. The quantum noiseprocess analysis device according to claim 15, wherein to predict thestate evolution of the quantum noise process within the subsequent timeaccording to the TTMs at the K time points comprises: calculate aquantum state ρ(t_(n)) of the quantum noise process at an n^(th) timepoint t_(n) according to the following formula:${{\rho \left( t_{n} \right)} = {\sum\limits_{m = 1}^{n - 1}{T_{m}{\rho \left( t_{n - m} \right)}}}},$wherein T_(m) represents a TTM at an m^(th) time point, and ρ(t_(n−m))represents a quantum state at an (n−m)^(th) time point t_(n−m), both nand m being positive integers.
 17. The method quantum noise processanalysis device according to claim 11, wherein to analyze the quantumnoise process according to the TTMs comprises: extract a second-ordermemory kernel of the quantum noise process according to the TTMs whenthe quantum noise process is a steady noise; calculate a correlationfunction of the quantum noise process according to the second-ordermemory kernel of the quantum noise process; and perform a Fouriertransform on the correlation function of the quantum noise process toobtain a frequency spectrum of the quantum noise process.
 18. The methodquantum noise process analysis device according to claim 17, wherein toextract the second-order memory kernel of the quantum noise processaccording to the TTMs at the K time points comprises: select N differentparameters, performing an experiment on the quantum noise process, andextracting memory kernels respectively corresponding to the N differentparameters from the experiment; and perform calculation according to thememory kernels respectively corresponding to the N different parameters,to obtain the second-order memory kernel of the quantum noise process.19. The method according to claim 17, wherein to calculate thecorrelation function of the quantum noise process according to thesecond-order memory kernel of the quantum noise process comprises:numerically extract the correlation function C_(αα′) of the quantumnoise process according to the following formula:${\underset{C_{{\alpha\alpha}^{\prime}}{(t_{n})}}{argmin}\left\{ {{{{\kappa_{2}\left( {t_{n};{C_{{\alpha\alpha}^{\prime}}\left( t_{n} \right)}} \right)} - {\kappa_{\exp}\left( t_{n} \right)}}} + {\left( {1 - \delta_{t_{n},t_{0}}} \right)\lambda_{n}{\sum\limits_{\alpha,\alpha^{\prime}}{{{C_{{\alpha\alpha}^{\prime}}\left( t_{n} \right)} - {C_{{\alpha\alpha}^{\prime}}\left( t_{n - 1} \right)}}}}}} \right\}},$wherein κ₂ represents the second-order memory kernel of the quantumnoise process, t_(n) represents the n^(th) time point, C_(αα′)(t_(n)) isa second-order correlation function at the n^(th) time point t_(n),κ_(exp) represents an approximate second-order memory kernel obtainedthrough an experiment, δ_(t) _(n) _(,t) ₀ is a Kronecker function, λ_(n)is an adjustable parameter, and C_(αα′)(t_(n−1)) is a second-ordercorrelation function at an (n−1)^(th) time point t_(n−1).
 20. Anon-transitory computer-readable medium for storing instructions,wherein the instructions when executed by a processor, cause theprocessor to: perform quantum process tomography (QPT) on a quantumnoise process of a target quantum system, to obtain dynamical maps ofthe quantum noise process wherein the QPT involves at least onemeasurement of the target quantum system; extract transfer tensor maps(TTMs) of the quantum noise process from the dynamical maps, the TTMsbeing used for representing a dynamical evolution of the quantum noiseprocess; and analyze the quantum noise process according to the TTMs.